UNIVERSITY OF BRISTOL Examination for the Degrees of B.Sc. and M.Sci. (Level 3) SET THEORY MATH 32000 (Paper Code MATH-32000) May/June 2016, 2 hours and 30 minutes This paper contains FOUR questions, all of which should be attempted for assessment. Calculators are not permitted in this examination. The Wellordering Principle may be assumed throughout this paper unless otherwise stated. On this examination, the marking scheme is indicative and is intended only as a guide to the relative weighting of the questions. Do not turn over until instructed. Page 1 of 3
Set Theory-16 Cont ... 1. (a) (9 marks) Give the definition of transitive set, the successor S(a) of any set a, inductive set, and countable set. (b) (8 marks) i. Give the von Neumann definition of natural number. ii. Prove carefully that every natural number is transitive. iii. Prove that w is transitive. (c) (8 marks) Determine the cardinality of a) the Hilbert Cube H = N[0, 1] = {f | f : N -> [0,1]}; b) the set of functions { F | F : H > H}, expressing them in the simplest form possible, and giving a careful reasoned argument in both cases .. 2. (a) (6 marks) State and prove the Principle of Transfinite Induction for On. (b) (13 marks) Prove the Recursion Theorem for Ordinals in the following form: Let F : V -> V be any function. Then there exists a unique function H : On -> V so that: Va( H(a) = F(H | a)). (c) (6 marks) Let the function F : On -> On be defined by the following recursion: F3(0) = No F(a +1) = 2F=(a) (cardinal exponentiation) Fy(X) = sup{F(x) | a < >} for > a limit ordinal. Prove that there is a fixed point for F3, i.e. an ordinal « with Fo(K) = K. Are such fixed points of F- always cardinals? Continued ... Page 2 of 3
Set Theory-16 Cont ... 3. (a) (9 marks) Define what it means: (i) for (A, R) to be a wellordering; (ii) for (X, E) to be an ordinal. (iii) State the Wellordering Principle, WP. (b) (8 marks) State the Axiom of Choice, (AC). Assume the Axiom of Choice and deduce the Wellordering Principle from it. (c) (8 marks) The Uniformisation Principle is stated as follows: If RC XxY is any relation, then there is a function f : X -> Y with (i) dom(f) = dom(R) =af {x|By((x, y) E R)}, and (ii) f & R. Prove the Uniformisation Principle from the Wellordering Principle. 4. (a) (6 marks) i. Give a definition of the Va-hierarchy and the wellfounded hierarchy of sets WF. ii. Define the rank function p(x) for any set x E WF. (b) (8 marks) Prove (i) a union of transitive sets is transitive; (ii) that if x is transitive then P(x) is transitive. Deduce that each Va is transitive. (c) (11 marks) If p(x) = p(y) = a compute (i) p({x,y}); (ii) p((x,y)) ; (iii) p((a+w)